Armendariz Extensions And Armendariz Factor Rings

A ring is called reduced if it has no nonzero nilpotent elements.Armendariz noted that a reduced ring R has the following property:for any polynomials f(x)= a0+a1x+…+amxm and g(x)= b0+b1x + …+bnxn over a ring R,f(x)g(x)= 0 0 implies aibi=0,0?i?m,0?j?n.Motivated by this fact,Rege and Chhawchharia named a ring with the above property Armen-dariz and initiated the research on Armendariz rings.We call the above-mentioned condition as Armendariz condition.Anderson and Camillo characterized Gaussian rings in terms of Armendariz rings.Hirano characterized the relation between the left annihilators of R and the left annihilators of R[x]by Armendariz rings.From the definition of Armendariz ring,we see that subrings of an Armendariz ring are Armendariz,but quotient rings of an Armendariz ring may not be Armendariz.So two natural questions arise:1.Which extension rings of Armendariz rings are Armendariz?2.Which quotient rings of Armendariz rings are Armendariz?In this thesis,we discuss the Armendariz property of group rings,trivial extensions,subrings of matrix rings,the factor rings of polynomial algebras modulo monomial ideals.In the second chapter,we discuss the Armendariz property of group algebras over a field and general group rings.We prove that in most cases,a group algebra is Armendariz if and only if it is reduced,which means that to characterize an Armendariz group algebra is equivalent to discussing the zero divisor conjecture of group algebras.The zero divisor conjecture says:a group algebra of a torsion-free group over a field is reduced.It is a famous difficult problem in group ring theory.For general group rings,our research indicates that group rings of cyclic groups and quaternion rings are crucial in the research of group rings.We also discuss the Armendariz property of the group ring over Hamilton's quaternion division ring.Let K8 denote the quaternion group of order 8.Theorem 2.2.12.Let R be a 3-torsion-free reduced commutative ring.Then the following conditions are equivalent:1.RK8 is Armendariz;2.RK8 is reduced;3.x2+ y2+ z2 = 0 has no nonzero solution in R.Theorerm 2.2.21.Let F be a field and G be a torsion or torsion-free group.Then FG is Armendariz if and only if either FG is reduced,or chF = p>0 and G is a torsion abelian group whose p-component is cyclic or quasi-cyclic.Theorerm 2.2.22.Let G be a group.1.The complex group algebra CG over G is Armendariz if and only if each group algebra over G of characteristic 0 is Armendariz.In this case,the torsion subgroup T of G is Abelian.2.Each group algebra over G is Armendariz if and only if each group algebra over G and any finite field is Armendariz.Let T be the subset of G consisting of all elements of finite order in G.Assume T is a subgroup of G.Write ?(G,T)= RG(1-T).Theorem 2.3.5.Suppose R is 2-torsion-free.Then RK8 is Armendariz if and only if the quaternion ring H(R)is Armendariz.Theorerm 2.3.8.If RG is an Armendariz ring,then1.T is a subgroup of G;2.?(G,T)is an Armendariz ring;3.R(G/T)is an Armendariz ring.Conversely,if the conditions above hold and ?T? is infinite or ?T? is finite and i-nvertible in R,then RG is an Armendariz ring.Let H denote Hamilton's quaternion division ring.Theorem 2.4.5.Let T be a torsion group.The group ring HT is Armendariz if and only if T is an elementary Abelian 2-group.Theorem 2.4.7.Let n be a positivc integer and g E H.Then H[x]/(Xn + q)is Armendariz if and only if one of the following conditions holds:(1)q = 0;(2)q(?)R;(3)n = 1;(4)n = 2,g G R and g<0.Theorem 2.4.9.Let f(x)be a polynomial with real coefficients and deg f = n ? 1.Then H[x]/(f(x))is Armenda-riz if and only if f(x)has n roots in R(counting multiplicities).In the third chapter,we discuss the Armendariz property of subrings of generalized matrix(especially,subrings of the trivial extension).We give the construction of subrings of the trivial extension,and give sufficient conditions and necessary conditions for subrings of the trivial extension to be Armendariz,which generalize the existing results.Theorerm 3.1.5.Let M be an R-bimodule.Given a sub-bimodzule K of M and derivative d:R?M/K,letThen Td,K is a subring of the trivial extension R ? M and ?(Td,k)=R.Conversely,every subring T of R ? M with ?(T)= R can be constructed in this way.Let ?:A ? B be a map between rings(not necessarily a ring homomorphism).If for each pair b1,b2 ? B with b1b2 = 0,there exist a1,a2 ? A such that ?(?1)= b1,?(a2)= b2,a2 = 0,then called ? reserving zero-product.Theorem 3.2.2.Let ?:A[x]?B[x]be a reserving zero-product ring homomorphism such that?(A)(?)B and ?(x)= x.If A is Armendariz,then B is Armendariz.Theorem 3.2.3.Let A,B be rings and let B be an Armendariz ring.If a map ?:A ? B reserves zero-product,then the extension map ?:A[x]?B[x]also reserves zero-product.Theorem 3.2.7.Let T be a subing of R? M,and ?(T)= R.1.If T is Armendariz,and Annr(Ann,(M0))= MO,then R is Armendariz;2.If T is Armendariz and ??T reserves zero-product,then ?fMg?f(-g)? = 1 for all f,g ? R[x]with f g = 0.3.If R is an Armendariz ring,M is an Armendariz bimodule,and fMg ?Mf(-g)=0 for all f,g ?R[x]with fg = 0,then T is an Armendariz ring.Theorem 3.2.8.Let T be a subring of R ? M such that ?(T)= R and ??T reserves zero-product.Suppose R is an Armendariz ring and M is an Armendariz bimodule.If T is Armendariz,then R ? M0 is Arme?dariz.Let M be an R-bimodule,?,T endomorphisms of R and a? =?(a)for a ? R.Set R ? M =R x M with multiplication defined as followsThen R ?T? M is a ring with identity.We present a sufficient and necessary condition for R ?T? M to be Armendariz.Theorem 3.3.2.Let R be a ring,M an R-bimodule,and ?,T endomorphisms of R.Then a sufficient and necessary condition for R ?T? M to be an Armendariz ring is.R is an Armendariz ring;2.M is an Armendariz(R?,RT)-bimodule;Theorem 3.3.5.Let ?:R ? A and T:R ? B be epimorphisms of rings,and let M be an(A,B)-bimodule that is viewed as an R-bimodule via the restriction of scalars.Let Then T(R,?,T,M)is an Armendariz ring if and only if R ? M is an Armendariz ring,in which R = R/(ker ?? ker T).Let ? be an endomorphism of R.Then ? is called compatible if a?(b)= 0 is equivalent to ab = 0 for all a,b ? R.For a positive integer n>2,letwhere k =[n/2],which means n = 2k when n is even and n = 2k + 1 when n is odd.Theorem 3.4.6.If R is a reduced ri'rng witlh compatible erndomorphisms ?1,?2,...,and Sn(R)denotes the subring of the matrix ring obtained by imposing ?1,?2,...,?n on diagonal elements of Un(R)in sequence.Then Sn(R)is Armendariz,In the forth chapter,we discuss the Armendariz property of K[x1,...,Xd]/I,in which K is a field and I is a monomial ideal.Let I be an ideal of R.If the factor ring R/I is Armendariz,then we say that I is an Armendariz ideal,briefly an A-ideal.Theorem 4.2.10.Let I and J be irreducible monomial ideals of R.Suppose I(?)J,J(?)I,and I and J are not all A-ideals.Then I? J is an A-ideal if and only if one of the following conditions holds.In the following theorems,the numbers under the braces denote the orders of the monomials.Theorem 4.3.1.Let k be a nonnegative integer.Then an ideal I is an A-ideal if its minimal generating set G(I)is one of the following three situationsTheorem 4.3.2.If the minimal generating set of I isthen I is an A-ideal.Theorem 4.3.4.Let c>2 and k,l ? 0,Then I is an A-ideal if the minimal generating set of I is one of the following situationsTheorem 4.3.5.Let c>1 and q>3.Then I is an A-ideal if the mi-nimal generating set of I isTheorem 4.4.6.Let I be A-ideal,then G(xiyi,xi1yi1)(?)C(I)for all xiyi,xi1,yi1 ?I with min{?i—i'?,?j-j'?}?6.